Study of BSVIEs was begun with the work of Lin in 2002 ([1]}). For general theory of BSVIEs, the readers are referred to Yong [2]}, [3]}, Wang–Zhang [4]}, Shi–Wang [5]}, Djordjević–Janković [6]}, Hu–Oksendal [7]}, Popier [8]} for the well-posedness of BSVIEs, to Wang–Yong [9]} for comparison theorems, to Shi–Wang–Yong [10]} for some extensions to mean-field BSVIEs, to Shi–Wen–Xiong [11]} for doubly BSVIEs, to Hamaguchi [12]} for infinite horizonal BSVIEs, to Wang [13]} for extended BSVIEs, to Wang–Yong [14]}, Hernández [15]}, Hernández–Possamai [16]} for BSVIEs with \(Z(s,s)\)  appears, to Agram–Djehiche [17]} for reflected BSVIEs, to Yong [18]}, Wang–Yong [19]}, Wang [13]}, Wang–Yong–Zhang [21]} for connections with PDEs, to Yong [2]}, [3]}, Wang–Zhang [24]}, Wang [25]} for applications in stochastic optimal control theory, to Yong [26]}, Wang–Shi [27]}, Agram [28]}, Beissner–Rosazza Gianin [29]}, Wang–Sun–Yong [30]} for connections with mathematical finance, and to Wang–Yong [14]}, Hamaguchi [32]} for applications to time-inconsistent stochastic optimal controls. It is well-known by now that the well-posedness can be established under the uniform Lipschitz conditions of \((y,z)\mapsto g(t,s,y,z)\)  for the general multi-dimensional cases. For the case of \(z\mapsto g(t,s,y,z)\)  being quadratic, the one-dimensional well-posedness was established by Wang–Sun–Yong [30]} with bounded free terms, and the multi-dimensional case was studied in Hernández [15]} with essentially small bounded free terms. To our best knowledge, it is completely blank for the general multi-dimensional super-linear BSVIEs (including sub-quadratic, strictly quadratic, and super-quadratic growth of \(z\mapsto g(t,s,y,z)\) ) with bounded/unbounded free terms. From mathematical point of view, it will be good to fill these blanks for BSVIEs, and to explore further, even for BSDEs. This is a major motivation of the current paper. Our main results can be briefly summarized as follows. Let us emphasis that all results are for multi-dimensional BSVIEs. Also, although we only consider the case of one-dimensional Brwonian motion, all the results hold for multi-dimensional Brownian motions, just with some modification of notations.
